Fixed point location for functions

Question

How are fixed points calculated?

Are intersections of $ y = f(x) ,y = f^{-1} (x) $ graphs give real fixed points for all $f$ ?

Answer 1

A fixed point is some $x$ such that $f(x) = x$. So, if you want to find a fixed point of a function, you need to intersect $y=f(x)$ and $y=x$.

If you intersect $y=f(x)$ and $y=f^{-1}(x)$ you can find some non-fixed points. As an example, consider $f(x) = \frac{1}{x}$, which is a function which coincides with its inverse function. The intersection of $y=f(x)$ and $y=f^{-1}(x)$ is the whole domain, while the only fixed points are $1$ and $-1$.